let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite)
what should i use here to prove the integral exist ???can someone give me the detail expanlation???

let f(x,t)=xe^(-xt).show that the integral I(x)=∫f(x,t)dt (integration from 0 to infinite)exists for all x>=0 . is x->I(x) continuous on [0,infinite) what should i use here to prove the integral exist ???can someone give me the detail expanlation???

thank you,then the limit is 1 so the integral exists,but how about the continuous part, for now x->I(x) ,I(X)=1 it seems ,it is continuous?? i am not sure

thank you,then the limit is 1 so the integral exists,but how about the continuous part, for now x->I(x) ,I(X)=1 it seems ,it is continuous?? i am not sure

Frankly, I have the same question about \(\displaystyle I(x)\).
We know that \(\displaystyle I(x)\ge 0\). But I have no idea how it fits into the question.

Isn't \(\displaystyle I(x)=1\) for \(\displaystyle x>0\) and \(\displaystyle I(x)=0\) for \(\displaystyle x=0\)? So it would be continuous on \(\displaystyle (0,\infty)\) and discontinuous at 0, right?